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Refined Gauss--Lobatto bounds for odd-order convexity

Tomasz Kania, Szymon Wąsowicz

arXiv:2609.34918v1math.NAcs.NAmath.CA

TL;DR

Classical Gauss and Gauss–Lobatto bounds bracket integrals of higher-order convex functions, but the odd-order bracket can be sharpened. The paper uses Peano-kernel dominance to construct a certified estimator with a sharp 1/4 error factor, and identifies an even-order obstruction from an odd, sign-changing Radau kernel.

  • Problem

    Classical Gaussian and Gauss–Lobatto bounds bracket integrals of higher-order convex functions, while relevant integrands may be nonsmooth or have nearby poles that make smoothness-based error constants unavailable or pessimistic.

    Source: §1 · p. 1, §8 · p. 13

  • Method

    The paper compares Gauss and Gauss–Lobatto Peano kernels to refine the odd-order bracket and form a certified estimator from the bracket’s centre.

    Source: §4.3 · p. 6

  • Results

    The estimator’s error factor 1/4 is sharp, and the even-order Radau midpoint is not consistently one-sided because its kernel is odd and sign-changing.

    Source: §1 · p. 2

  • Takeaways & Limitations

    The refined certificate applies to higher-order shape-constrained integration, including truncated powers and near-pole Stieltjes kernels.

    Source: §1 · p. 1, §8 · p. 13

    The refined half-bracket is established for the Legendre pair; analogous dominance for other weights, including Jacobi weights, is not claimed.

    Source: §4.3 · p. 6

Abstract

from arXiv · show

For functions that are convex of higher order, classical extremalities identify Gaussian quadrature as a lower bound and Gauss--Lobatto quadrature as an upper bound for the integral. We sharpen this bracket in every odd order: if $f$ is $(2n-1)$-convex on $[-1,1]$, then the exact integral $I[f]:=\int_{-1}^{1}f(x)\,dx$ lies not merely between the $n$-point Gauss--Legendre rule $G_n[f]$ and the $(n+1)$-point Gauss--Lobatto rule $L_{n+1}[f]$, but already between $G_n[f]$ and their midpoint. This yields the certified estimator $Q_n[f]:=\frac{3}{4}G_n[f]+\frac{1}{4}L_{n+1}[f]$, with $|I[f]-Q_n[f]|\leqslant\frac{1}{4}|L_{n+1}[f]-G_n[f]|$, and the constant $\frac{1}{4}$ is sharp. We also record a complementary phenomenon in even order: for $2n$-convex functions the Gauss--Radau endpoint rules still bracket the integral, but the midpoint of the two Radau rules cannot yield a one-sided refinement. The obstruction is explained by an odd, sign-changing Radau quadrature kernel. Finally, we distinguish numerical certification from ordinary high-order approximation: we include concrete examples from spline, moment and statistical models and report experiments for truncated powers and near-pole Stieltjes kernels, where the shape certificate remains meaningful when classical smoothness-based error constants are unavailable or severely pessimistic.

1. Introduction

The paper sharpens the odd-order Gauss–Lobatto bracket and explains why the analogous even-order midpoint refinement fails. Its focus is certified integration under higher-order shape constraints, including integrands that may not be analytic.

1. Introduction

  • Higher-order convexity arises in spline approximation, moment problems, and shape-constrained statistics, including settings with piecewise-polynomial or near-pole integrands.Examples include truncated powers, Stieltjes–Markov transforms, and integrated spline processes.Source: §1 · p. 1
  • The odd-order refinement places the integral closer to the Gauss side, and the estimator’s constant 1/4 is sharp.Taking the refined interval’s centre gives the estimator; the reported bound is expressed relative to the Gauss–Lobatto gap.Source: §1 · p. 2
  • In even order, Gauss–Radau endpoint rules still bracket the integral, but their midpoint is not consistently one-sided because the associated Peano kernel is odd and sign-changing.The paper relates this obstruction to the impossibility of symmetry for a definite quadrature formula of odd Peano order.Source: §1 · p. 2
  • The numerical discussion distinguishes certification from smooth-function approximation and focuses on truncated powers and near-pole Stieltjes kernels where smoothness-based error constants may be unavailable or pessimistic.The analytic tests 1/x, ex, and log(1 + x) are retained as sanity checks in the companion repository.Source: §1 · p. 2
  • The paper treats Gauss–Legendre and Gauss–Lobatto rules as certification devices, with the refined bracket relying on pointwise Peano-kernel dominance.The authors are not aware of an analogous dominance principle for Clenshaw–Curtis-type or Szegő–Lobatto rules.Source: §1 · p. 2

2. Higher-order convexity

Higher-order convexity is formulated through divided differences and connected to quadrature inequalities through a representation by truncated powers. This representation reduces functional inequalities to tests on a one-parameter family.

2.1. Definitions and basic properties.

  • Higher-order convexity is defined by nonnegative divided differences, recovering ordinary convexity at order one and, for sufficiently smooth functions, following from a nonnegative highest derivative.The definition is on [-1, 1]; the paper also gives a recursive characterization through monotonicity of repeated-argument divided differences.Source: §2.1 · pp. 2–3

2.2. Popoviciu-type representation.

  • The cone of higher-order convex functions is generated by polynomials of degree at most m − 1 and nonnegative mixtures of truncated powers.The representation also applies without a smoothness assumption, using a nonnegative Borel measure; for concave functions the measure is nonpositive.Source: §2 · p. 2, §2.2 · p. 3
  • The representation reduces a linear-functional inequality to checking it on truncated powers, a strategy used for differences of quadrature operators.The paper identifies this representation as the bridge between higher-order convexity and Peano kernels.Source: §2.2 · pp. 3–4

3. Gauss-type quadratures on [−1, 1]

The paper sets out the Legendre Gauss-type rules and their common structural properties for later Peano-kernel comparisons. It also explains how the results transfer under affine changes of variables and how convexity responds to orientation reversal.

3.1. The integral operator and three families of rules.

  • The Gauss–Legendre and Gauss–Lobatto rules have positive weights and share precision 2n − 1; their common Gauss-type structure underlies interlacing and the Peano-kernel comparison.The (n + 1)-point Lobatto rule prescribes both endpoints, while the Gauss–Legendre rule uses the zeros of the Legendre polynomial.Source: §3.1 · p. 4

3.2. Affine changes of variables.

  • The quadrature inequalities on [-1, 1] transfer to any finite interval by affine rescaling of nodes and weights.The weights are multiplied by (b − a)/2.Source: §3.2 · pp. 4–5
  • Under affine transformations, odd-order convexity is preserved when the map reverses orientation, whereas even-order convexity changes to concavity.The paper’s focus on odd-order convexity means no increasing-versus-decreasing distinction is needed in its sequel.Source: §3.2 · p. 5

4. Odd order: a refined Gauss–Lobatto inequality

For odd-order convexity, the paper refines the classical Gauss–Lobatto bracket by localizing the integral closer to the Gauss side. It establishes this refinement through a comparison of Peano kernels and identifies the Legendre-specific scope of that comparison.

4.1. Classical extremalities.

  • The integral is localized closer to the Gauss rule within the classical Gauss–Lobatto bracket for odd-order convex functions.The paper presents this as a refinement of the classical extremalities, not as a new extremality result.Source: §4.1 · p. 5

4.2. Peano kernels for the two error functionals.

  • The Gauss and Lobatto error functionals have Peano representations of the same order, with non-negative kernels.These representations provide the kernel framework for comparing the two errors.Source: §4.2 · p. 5

4.3. Kernel dominance and the refined bracket.

  • For the Legendre Gauss/Gauss–Lobatto pair, kernel dominance gives the refined bracket; the proof invokes Brass–Schmeisser [7].The rules share degree of precision 2n−1, and the kernels have the orientations KG ≥ 0 for I−Gn and KL ≥ 0 for Ln+1−I.Source: §4.3 · p. 6
  • The half-bracket extension is not claimed for other weights; it requires a corresponding kernel-domination theorem.The stated scope is the Legendre principal pair.Source: §4.3 · p. 6
  • For (2n−1)-concave functions, the refined inequalities reverse.The paper obtains this case by applying the convex result to −f.Source: §4.3 · p. 7

5. Certified estimators and optimality

The paper turns the refined interval into a certified estimator, establishes two distinct forms of optimality, and extends the error bound to composite integration. The resulting local bounds can support stopping and refinement decisions.

5.1. A sharp certified one-point estimate.

  • The estimator Qn is the centre of the certified refined interval, with an a posteriori error bound of 1/4 |Ln+1[f] − Gn[f]|.The refined interval is [Gn[f], Mn[f]], where Mn[f] is the midpoint of the Gauss and Lobatto values.Source: §5.1 · p. 7, §5 · p. 7

5.2. Optimality of the interval centre.

  • Among affine combinations Aθ, θ = 1/4 uniquely minimizes the worst-case distance to the certified interval, with radius 1/4 |Ln+1[f] − Gn[f]|.This is interval-theoretic optimality: the best affine decision given only the certified interval is its centre.Source: §5.2 · p. 8

5.3. Function-theoretic sharpness of the constant.

  • The error-bound constant 1/4 is sharp on the cone of (2n−1)-convex functions.Endpoint-concentrating truncated powers provide a sequence approaching sharpness; for that construction, Gn[fε] = 0 and Ln+1[fε] = vn ε^(2n−1), with vn = 2/(n(n + 1)).Source: §5.3 · pp. 8–9

5.4. Composite rules and stopping criteria.

  • The one-interval estimate extends to composite integration by summing local bounds for (2n−1)-convex or concave functions.For the concave case, the local gap is replaced by its absolute value.Source: §5.4 · p. 9, Then on each subinterval · p. 9

Then on each subinterval

  • The composite estimate supplies a stopping criterion, and the companion code implements uniform refinement and greedy bisection.Greedy bisection selects the subinterval with the largest local contribution.Source: Then on each subinterval · p. 9

6. The smooth and analytic regime

The smooth-regime discussion distinguishes certification from classical approximation and examines the estimator’s leading error constant, a degree-raising combination, and analytic error bounds. Analytic estimates can be sharp when a trustworthy domain of analyticity is available, while the paper’s certified inequalities require no analyticity.

6. The smooth and analytic regime

  • The smooth-regime analysis asks how the certified estimator behaves on smooth integrands, where classical error expansions can also support degree-raising combinations.The discussion separates a rigorous a posteriori bound from the estimator’s leading error constant.Source: §6 · p. 9, §6.1 · p. 10

6.1. Error constants and a degree-raising combination.

  • Q_n has the same formal order as G_n and L_{n+1}, but its principal error constant is smaller than G_n’s by the factor (2n −1)/(4n).The discussion identifies this leading constant on the first non-exact monomial.Source: §6.1 · pp. 10–11
  • P_n combines G_n and L_{n+1} to cancel their leading x2n errors, making it exact on Π2n+1 but not certified for (2n −1)-convex functions.Unlike Q_n, P_n is not constrained to the refined interval.Source: §6.1 · p. 11

6.2. Analytic functions and complex error kernels.

  • Analytic error estimates use complex Peano kernels and can be sharp when a trustworthy analyticity domain is available, whereas the paper’s certified inequalities require no analyticity.The cited work studies maxima of the complex kernel for symmetric Gauss–Lobatto formulas.Source: §6.2 · p. 11

7. Even order: Radau quadrature kernels and an obstruction

For 2n-convex functions, the two Gauss–Radau endpoint rules bracket the integral. Their midpoint cannot give a one-sided refinement: its odd, nonzero Peano kernel changes sign.

7.1. The Radau bracket for 2n-convexity.

  • The two Gauss–Radau endpoint rules bracket the integral for 2n-convex functions, with the inequalities reversed for 2n-concave functions.The bracket follows from the non-negative Peano kernels of the endpoint rules.Source: §7.1 · p. 12

7.2. Why the Radau midpoint cannot refine the bracket.

  • For every n ≥ 1, the Radau midpoint kernel is odd and nonzero, so it changes sign and precludes a universal one-sided refinement.The sign change is established by testing the kernel near the endpoints.Source: §7.2 · p. 12
  • The obstruction is structural: a definite quadrature formula of odd Peano order cannot be symmetric.The kernel argument establishes this obstruction for all n.Source: §7.2 · p. 13

8. Numerical experiments

The experiments assess certified integration under higher-order shape constraints, rather than competing with high-order quadrature on smooth functions with known exact answers. They include a truncated-power spline diagnostic, near-pole Stieltjes kernels, and plots of the Gauss/Lobatto and Radau kernels.

8. Numerical experiments

  • The numerical study treats certification as distinct from ordinary high-order approximation and focuses on truncated powers and near-pole Stieltjes kernels.The smooth tests are retained only as sanity checks in the companion repository.Source: §8 · p. 13

8.1. A spline-basis diagnostic.

  • At tolerance 10−8, greedy certified Q4 reaches a bound of 2.382×10−9 and actual error of 1.614×10−9 using two subintervals and 23 distinct evaluations on the truncated-power diagnostic.For n = 4, the example is 7-convex but not C8, so the usual bounded-eighth-derivative Gauss error constant is unavailable.Source: §8.1 · p. 13

8.2. Near-pole Stieltjes kernels and adaptive cost.

  • For δ approaching the interval, certified Q4 stays within a small factor of true error and uses substantially fewer evaluations than adaptive Simpson for the three reported δ values, while Simpson has smaller true error without a shape certificate.Both methods refine near the left endpoint; the comparison uses absolute tolerance 10−8.Source: §8.2 · p. 14, Table 1 · p. 14

8.3. Kernel plots.

  • For n = 4, Figure 1 illustrates 0 ≤ KG ≤ KL; the Radau kernel plot illustrates oddness and sign change, which obstruct a one-sided even-order refinement.Figure 1 also plots the kernels’ half-difference.Source: §8.3 · pp. 14–15, Figure 1 · p. 15, Figure 2 · p. 15
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